Q.What is the 21st term of the sequence defined by $a_n = (n-1)(20-n)(3+n)$?

Sequence Term Evaluation

Picture a staircase where the height of each step follows a rule: step 1 is 1 unit high, step 2 is 4 units, step 3 is 9 units, step 4 is 16 units. The pattern is height = (step number)$^2$. If someone asks for the height of step 20, you don't climb up counting -- you compute $20^2 = 400$ directly. That single act of substitution is what sequence term evaluation means.

The Core Idea

A sequence is an ordered list of numbers, and each number is called a term. Its position -- first, second, third, ... -- is the index, usually written $n$ (starting at $n=1$ unless told otherwise). The term at position $n$ is written $a_n$.

Think of the sequence as a machine: feed it an index $n$, and it returns the term $a_n$.

• Input $n=1$ -> output $a_1 = 1^2 = 1$
• Input $n=2$ -> output $a_2 = 2^2 = 4$
• Input $n=10$ -> output $a_{10} = 10^2 = 100$

You are not solving anything here -- you are purely substituting a number into a formula and simplifying.

The Precise Statement

A sequence defined this way is a function whose domain is the positive integers. Evaluating a term means computing $f(n)$ for one specific value of $n$.

Example 1 -- sequence $a_n = 3n + 2$:

Example 2 -- sequence $a_n = \dfrac{(-1)^n}{n}$:

Quick Check

Given $a_n = \dfrac{n^2 - 1}{n+1}$, find $a_4$.

$a_4 = \frac{4^2 - 1}{4 + 1} = \frac{16-1}{5} = \frac{15}{5} = 3$ …